{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:V4HLYNOMM3RBBNUYRGPZ7NVFGQ","short_pith_number":"pith:V4HLYNOM","canonical_record":{"source":{"id":"1604.07979","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-27T08:49:36Z","cross_cats_sorted":[],"title_canon_sha256":"ee2e53eadca6954c8e275b03b196ed659d4180072fd654b9046cf943c781266e","abstract_canon_sha256":"00651b76f39d6af83e8a3f0c1a690907a3336179e0dedf28e7894296e4352b13"},"schema_version":"1.0"},"canonical_sha256":"af0ebc35cc66e210b698899f9fb6a5343b85ed4aa6e15c144146720367f64450","source":{"kind":"arxiv","id":"1604.07979","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07979","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07979v1","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07979","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"pith_short_12","alias_value":"V4HLYNOMM3RB","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4HLYNOMM3RBBNUY","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4HLYNOM","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:V4HLYNOMM3RBBNUYRGPZ7NVFGQ","target":"record","payload":{"canonical_record":{"source":{"id":"1604.07979","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-27T08:49:36Z","cross_cats_sorted":[],"title_canon_sha256":"ee2e53eadca6954c8e275b03b196ed659d4180072fd654b9046cf943c781266e","abstract_canon_sha256":"00651b76f39d6af83e8a3f0c1a690907a3336179e0dedf28e7894296e4352b13"},"schema_version":"1.0"},"canonical_sha256":"af0ebc35cc66e210b698899f9fb6a5343b85ed4aa6e15c144146720367f64450","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:08.485954Z","signature_b64":"CfoF3ltemRCNzfBB2L/4Bpcmdc6FrrP38a7fstEWe4i3GlgJIFEQKh7iYLNNkUxzY4MKYsAURFO1YhC4WqVvCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af0ebc35cc66e210b698899f9fb6a5343b85ed4aa6e15c144146720367f64450","last_reissued_at":"2026-05-18T01:16:08.485261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:08.485261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.07979","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:16:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HUjZiq6VNUGWKOEPpGiuGImqC5oHN5mjstIt/LFj3SlS/Yw7SxV6K1JAwHI+9/+eWJCRjbmMAYTzZKjlyCmyBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:20:11.253622Z"},"content_sha256":"19c25c9e1aa510af3a11f1a2c95d2e69581b3588b5d06bdda3e06fcd3656ee60","schema_version":"1.0","event_id":"sha256:19c25c9e1aa510af3a11f1a2c95d2e69581b3588b5d06bdda3e06fcd3656ee60"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:V4HLYNOMM3RBBNUYRGPZ7NVFGQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boundedness and closedness of linear relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Guixin Xu, Guojing Ren, Yuming Shi","submitted_at":"2016-04-27T08:49:36Z","abstract_excerpt":"This paper studies boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness, and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07979","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:16:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tDVmnU8PFkM6yuLd2IC9IL9ideTFIezzuQYTozDwhB5FU6pVOicL26ofnp9O21ckrNMUUJpkPNJhpURCTbSmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T15:20:11.254128Z"},"content_sha256":"d9c850fff2b5474c169a5762dd93dcb864dfeced248cf4ae05c5109f304a2465","schema_version":"1.0","event_id":"sha256:d9c850fff2b5474c169a5762dd93dcb864dfeced248cf4ae05c5109f304a2465"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/bundle.json","state_url":"https://pith.science/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T15:20:11Z","links":{"resolver":"https://pith.science/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ","bundle":"https://pith.science/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/bundle.json","state":"https://pith.science/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4HLYNOMM3RBBNUYRGPZ7NVFGQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:V4HLYNOMM3RBBNUYRGPZ7NVFGQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"00651b76f39d6af83e8a3f0c1a690907a3336179e0dedf28e7894296e4352b13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-27T08:49:36Z","title_canon_sha256":"ee2e53eadca6954c8e275b03b196ed659d4180072fd654b9046cf943c781266e"},"schema_version":"1.0","source":{"id":"1604.07979","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07979","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07979v1","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07979","created_at":"2026-05-18T01:16:08Z"},{"alias_kind":"pith_short_12","alias_value":"V4HLYNOMM3RB","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4HLYNOMM3RBBNUY","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4HLYNOM","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:d9c850fff2b5474c169a5762dd93dcb864dfeced248cf4ae05c5109f304a2465","target":"graph","created_at":"2026-05-18T01:16:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper studies boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness, and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and ","authors_text":"Guixin Xu, Guojing Ren, Yuming Shi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-27T08:49:36Z","title":"Boundedness and closedness of linear relations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07979","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:19c25c9e1aa510af3a11f1a2c95d2e69581b3588b5d06bdda3e06fcd3656ee60","target":"record","created_at":"2026-05-18T01:16:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"00651b76f39d6af83e8a3f0c1a690907a3336179e0dedf28e7894296e4352b13","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-27T08:49:36Z","title_canon_sha256":"ee2e53eadca6954c8e275b03b196ed659d4180072fd654b9046cf943c781266e"},"schema_version":"1.0","source":{"id":"1604.07979","kind":"arxiv","version":1}},"canonical_sha256":"af0ebc35cc66e210b698899f9fb6a5343b85ed4aa6e15c144146720367f64450","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af0ebc35cc66e210b698899f9fb6a5343b85ed4aa6e15c144146720367f64450","first_computed_at":"2026-05-18T01:16:08.485261Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:08.485261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CfoF3ltemRCNzfBB2L/4Bpcmdc6FrrP38a7fstEWe4i3GlgJIFEQKh7iYLNNkUxzY4MKYsAURFO1YhC4WqVvCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:08.485954Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.07979","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19c25c9e1aa510af3a11f1a2c95d2e69581b3588b5d06bdda3e06fcd3656ee60","sha256:d9c850fff2b5474c169a5762dd93dcb864dfeced248cf4ae05c5109f304a2465"],"state_sha256":"b50ef691d2f07937dc0105da2c95fc4e211fda2860f1677d6d5c88c552fc2e69"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OetHudV42dvBAWd6DNXmjPNvvD7plX+ajkEHfRTOkJ6yDvdniWlETp3VJWx1fYIXz5MKJWAfEuFpyyQw8DNUBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T15:20:11.256939Z","bundle_sha256":"18c1e62e6651adb7e2b1af78cd276484b5d6d47098858cf0dcbfbb6ef0f5189f"}}